/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 The set \(S=\left\\{e^{3 t}, t e... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 8

The set \(S=\left\\{e^{3 t}, t e^{3 t}, t^{2} e^{3 t}\right\\}\) is a basis of a vector space \(V\) of functions \(f: \mathbf{R} \rightarrow \mathbf{R}\). Let \(\mathbf{D}\) be the differential operator on \(V ;\) that is, $\mathbf{D}(f)=d f / d t\(. Find the matrix representation of \)\mathbf{D}$ relative to the basis \(S\). Find the image of each basis function: \\[ \begin{array}{ll} \mathbf{D}\left(e^{3 r}\right) & =3 e^{3 t} \\ \mathbf{D}\left(t e^{3 t}\right) & =e^{3 t}+3 t e^{3 t} \end{array}=3\left(e^{3 t}\right)+0\left(t e^{3 t}\right)+0\left(t^{2} e^{3 t}\right) \\] \\[ [\mathbf{D}]=\left[\begin{array}{lll} 3 & 1 & 0 \\ 0 & 3 & 2 \\ 0 & 0 & 3 \end{array}\right] \\] \\[ \mathbf{D}\left(t^{2} e^{3 t}\right)=2 t e^{3 t}+3 t^{2} e^{3 t}=0\left(e^{3 t}\right)+2\left(t e^{3 t}\right)+3\left(t^{2} e^{3 t}\right) \\]

Short Answer

Expert verified
The matrix representation of the differential operator \(\mathbf{D}\) relative to the basis \(S\) is: \[[\mathbf{D}] = \left[\begin{array}{lll} 3 & 1 & 0 \\ 0 & 3 & 2 \\ 0 & 0 & 3 \end{array}\right]\]

Step by step solution

01

Find the image of the first basis function

First, find the image of \(e^{3t}\) under the operator D by taking the derivative: \[\mathbf{D}(e^{3t}) = \frac{d}{dt}(e^{3t}) = 3e^{3t}\]
02

Express the first image in terms of the basis functions

Next, we express the result from Step 1 in terms of the basis functions: \[\mathbf{D}(e^{3t}) = 3e^{3t} = 3\left(e^{3t}\right) + 0\left(te^{3t}\right) + 0\left(t^{2}e^{3t}\right)\]
03

Find the image of the second basis function

Repeat for the second basis function, \(te^{3t}\): \[\mathbf{D}(te^{3t}) = \frac{d}{dt}(te^{3t}) = e^{3t} + 3te^{3t}\]
04

Express the second image in terms of the basis functions

Express the result from Step 3 in terms of the basis functions: \[\mathbf{D}(te^{3t}) = e^{3t} + 3te^{3t} = 1\left(e^{3t}\right) + 3\left(te^{3t}\right) + 0\left(t^{2}e^{3t}\right)\]
05

Find the image of the third basis function

Again, for the third basis function, \(t^2e^{3t}\): \[\mathbf{D}(t^2e^{3t}) = \frac{d}{dt}(t^2e^{3t}) = 2te^{3t} + 3t^2e^{3t}\]
06

Express the third image in terms of the basis functions

Express the result from Step 5 in terms of the basis functions: \[\mathbf{D}(t^2e^{3t}) = 2te^{3t} + 3t^2e^{3t} = 0\left(e^{3t}\right) + 2\left(te^{3t}\right) + 3\left(t^{2}e^{3t}\right)\]
07

Construct the matrix representation of the differential operator

Finally, we can construct the matrix representation of the operator D by using the coefficients from Steps 2, 4, and 6 in the matrix: \[[\mathbf{D}] = \left[\begin{array}{lll} 3 & 1 & 0 \\ 0 & 3 & 2 \\ 0 & 0 & 3 \end{array}\right]\]

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Most popular questions from this chapter

How are the matrices \(A\) and \(B\) in Problem \(6.34\) related? By Theorem 6.12, the matrices \(A\) and \(B\) are equivalent to each other; that is, there exist nonsingulan matrices \(P\) and \(Q\) such that \(B=Q^{-1} A P\), where \(P\) is the change-of-basis matrix from \(S\) to \(E\), and \(Q\) is the change-of-basis matrix from \(E\) to \(S\). Thus, $$ P=\left[\begin{array}{ll} 1 & 2 \\ 3 & 5 \end{array}\right], \quad Q=\left[\begin{array}{rr} -5 & 2 \\ 3 & -1 \end{array}\right], \quad Q^{-1}=\left[\begin{array}{ll} 1 & 2 \\ 3 & 5 \end{array}\right] $$ and $$ Q^{-1} A P=\left[\begin{array}{ll} 1 & 2 \\ 3 & 5 \end{array}\right]\left[\begin{array}{rr} -8 & -23 \\ 5 & -13 \end{array}\right]\left[\begin{array}{ll} 1 & 2 \\ 3 & 5 \end{array}\right]=\left[\begin{array}{rr} -7 & -11 \\ 13 & 22 \end{array}\right]=B $$

Suppose \(S=\left\\{u_{1}, u_{2}\right\\}\) is a basis of \(V\), and \(T: V \rightarrow V\) is defined by \(T\left(u_{1}\right)=3 u_{1}-2 u_{2}\) and \(T\left(u_{2}\right)=u_{1}+4 u_{2}\) Suppose \(S^{\prime}=\left\\{w_{1}, w_{2}\right\\}\) is a basis of \(V\) for which \(w_{1}=u_{1}+u_{2}\) and \(w_{2}=2 u_{1}+3 u_{2}\) (a) Find the matrices \(A\) and \(B\) representing \(T\) relative to the bases \(S\) and \(S^{\prime}\), respectively. (b) Find the matrix \(P\) such that \(B=P^{-1} A P\).

Let \(A: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}\) be defined by the matrix \(A=\left[\begin{array}{rr}5 & -1 \\ 2 & 4\end{array}\right]\). (a) Find the matrix \(B\) representing \(A\) relative to the basis \(S=\left\\{u_{1}, u_{2}\right\\}=\\{(1,3),(2,8)\\}\). (Recall that \(A\). represents the mapping \(A\) relative to the usual basis \(E\).) (b) For \(v=(a, b)\), find \([v]_{S}\) and \([A(v)]_{s}\).

Consider the following \(3 \times 3\) matrix \(A\) and basis \(S\) of \(\mathbf{R}^{3}\) : \\[A=\left[\begin{array}{rrr}1 & -2 & 1 \\\3 & -1 & 0 \\\1 & 4 & -2 \end{array}\right] \quad \text { and } \quad S=\left\\{u_{1}, u_{2}, u_{3}\right\\}=\left\\{\left[\begin{array}{l}1 \\\1 \\\1 \end{array}\right], \quad\left[\begin{array}{l}0 \\\1 \\ 1\end{array}\right], \quad\left[\begin{array}{l}1 \\\2 \\\3\end{array}\right]\right\\}\\]

Consider the linear operator \(F\) on \(\mathbf{R}^{2}\) defined by \(F(x, y)=(5 x+y, 3 x-2 y)\) and the following bases of \(\mathbf{R}^{2}\) : $$ S=\\{(1,2),(2,3)\\} \quad \text { and } \quad S^{\prime}=\\{(1,3),(1,4)\\} $$ (a) Find the matrix \(A\) representing \(F\) relative to the basis \(S\). (b) Find the matrix \(B\) representing \(F\) relative to the basis \(S^{\prime}\). (c) Find the change-of-basis matrix \(P\) from \(S\) to \(S^{\prime}\). (d) How are \(A\) and \(B\) related?
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